In what way, do Distributions (Generalised Functions) generalise the notion of function?

Question

In what way does the concept of Generalised Function, or Distribution, generalise the notion of function?

Is it only because we can now define derivatives in the weak sense? or is it because some other property?

Answer 1

Okay. I'll give this a go. I think a good question to ask is: What do we mean by a function? In introductory real analysis, the formalism is that a function is a map $f:U\to \mathbb{C},$ where $U\subseteq \mathbb{R}^n$ is some appropriately nice subset, say closed or open. We usually impose some regularity on $f$ such as continuity.

Now, a function in the modern sense (post-measure theory) is really an almost everywhere equivalence class of measurable maps $f:U\to \mathbb{C}$. Why? Well, the insistence on viewing things as map sort of collapses under even small perturbations. What if I change the value of $f$ at a point? Well, it doesn't matter if $f$ were continuous - we could even recover the old value! Integrals don't change either.

So what happens? Well, as long as you're in the realm of continuous maps, these are exactly the maps whose exact values at points follow from their "general behaviour" - their almost everywhere class. On the other hand, what's the value of, say, the Heaviside function at $x=0$? Answer is: It doesn't really matter, right? The genuinely useful statement is that the Heaviside function is piecewise continous, equal to $1$ on $(0,\infty)$ and $0$ on $(-\infty,0)$ - the jump at $0$ is the essential information.

With this view of what a function is, it's not that far of a stretch to say that, say, Radon measures are functions as well. It's meaningful to say that $\delta_0$ is $0$ away from $0$. It's meaningful to talk about integrals of measures. In other words, measures do have meaningful local behaviours. As such, they naturally inhabit the same properties as the almost everywhere classes of maps that form the measure theoretic functions.

Now, of course, it's also lovely that the notion of weak derivatives allows one to expand the theoretical foundations of partial differential equations. However, the weak derivative of a function is always a measure (identifying any $L^1_{loc}$ function $f$ with the measure $f\textrm{d}x$). I think the one good reason to consider a general distribution a generalised function is that we then get the statement "the weak derivative of a generalised function is a generalised function", so that the distributions fit into a PDE framework, and hence are useful for doing theoretical physics.

So to sum up: I think it's somewhat natural to call a Radon measure a generalised function, simply thinking about what we actually want a function to be, and distributions in general are then labelled generalised functions because it makes the theory of weak derivatives work in general.

Answer 2

In what way does the concept of Generalised Function, or Distribution, generalise the notion of function?

It seems that Laurent Schwartz, who created the theory of distributions, give us an answer:

By a generalization, I mean that the set of distributions is a larger set than the set of functions; every function is a particular distribution, but there are distributions which are not functions. ([1], p. 211)

Is it only because we can now define derivatives in the weak sense? or is it because some other property?

I would say that, when talking about distributions as generalization of functions, there are a set of fundamental properties (which are summarized in the axiomatic theory) that we should be aware of ([2], p. 4-7):

• The set $\mathcal{D'}(I)$ of distributions on $I$ contains the set $C(I)$ of continuous functions on $I$: $$C(I)\subset \mathcal{D'}(I).$$
• The derivation in the distributional sense is an operator $D$ from $\mathcal{D'}(I)$ to $\mathcal{D'}(I)$ which extends the classical derivation on $C^1(I)$: $$\left\{\begin{aligned} &Df\in\mathcal{D'}(I),&&\forall\ f\in \mathcal{D'}(I)\\ &Df=f',&&\forall\ f\in C^1(I)\end{aligned}\right.$$
• $\mathcal{D'}(I)$ has no more elements than those which are needed to ensure that each continuous function is infinitely differentiable: $$\text{For all $f\in \mathcal{D'}(I)$, there exists $h\in C(I)$ such that $D^n h=f$ for some $n\in\mathbb N$.}$$
• Given $f,g\in C(I)$ and $n\in \mathbb N$, we have $Df=Dg$ if and only if $f-g$ is constant. In general: $$D^nf=D^ng\;\Longleftrightarrow\; f-g\text{ is a polynomial of degree less than $n$}$$

Essentially, these properties are the way $\mathcal{D'}(I)$ generalizes $C(I)$ and the reason why the generalization is useful.

[1] A Mathematician Grappling with His Century by Laurent Schwartz.

[2] Introduction to the Theory of Distributions by J. C. Ferreira, R F Hoskins and J. Sousa-Pinto.